A111949 - OEIS

%I #29 Nov 22 2018 21:50:11

%S 1,-1,-2,1,1,2,-2,-1,3,-1,0,-2,0,2,-2,1,0,-3,0,1,4,0,-2,2,1,0,-4,-2,2,

%T 2,0,-1,0,0,-2,3,0,0,0,-1,2,-4,-2,0,3,2,-2,-2,3,-1,0,0,0,4,0,2,0,-2,0,

%U -2,2,0,-6,1,0,0,-2,0,4,2,0,-3,0,0,-2,0,0,0,0,1,5,-2,-2,4,0,2,-4,0,2,-3,0,-2,0,2,0,2,0,-3,0,1,2,0,-2,0,4

%N Expansion of eta(q) * eta(q^2) * eta(q^10) * eta(q^20) / (eta(q^4) * eta(q^5)) in powers of q.

%C Number 37 of the 74 eta-quotients listed in Table I of Martin (1996).

%H Y. Martin, <a href="http://dx.doi.org/10.1090/S0002-9947-96-01743-6">Multiplicative eta-quotients</a>, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

%H Michael Somos, <a href="/A030203/a030203.txt">Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers</a>

%F Euler transform of period 20 sequence [-1, -2, -1, -1, 0, -2, -1, -1, -1, -2, -1, -1, -1, -2, 0, -1, -1, -2, -1, -2, ...].

%F a(n) is multiplicative with a(p^e) = (-1)^e if p = 2, a(p^e) = 1 if p = 5, a(p^e) = (1 + (-1)^e) / 2 if p == 11, 13, 17, 19 (mod 20), a(p^e) = e + 1 if p == 1, 9 (mod 20), a(p^e) = (e + 1)*(-1)^e if p == 3, 7 (mod 20).

%F G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 - x^k) * (1 - x^(2*k)) / (1 - x^(5*k)).

%F G.f.: Sum_{k>0} Kronecker(k, 5) * x^k / (1 + x^(2*k)).

%F G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(5*k)) * (1 - x^(20*k)) / (1 + x^(2*k)).

%F |a(n)| = A035170(n). a(2*n) = -a(n). a(2*n + 1) = A129391(n). a(4*n + 3) = -2 * A033764(n).

%F a(5*n) = a(n). - _Michael Somos_, May 19 2015

%e G.f. = q - q^2 - 2*q^3 + q^4 + q^5 + 2*q^6 - 2*q^7 - q^8 + 3*q^9 - q^10 + ...

%t a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^10] QPochhammer[ q^20] / (QPochhammer[ q^4] QPochhammer[ q^5]), {q, 0, n}]; (* _Michael Somos_, May 19 2015 *)

%t a[ n_] := If[ n < 1, 0, Sum[ Mod[d, 2] (-1)^Quotient[d, 2] KroneckerSymbol[ n/d, 5], { d, Divisors[ n]}]]; (* _Michael Somos_, May 19 2015 *)

%o (PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^10 + A) * eta(x^20 + A) / eta(x^4 + A) / eta(x^5 + A), n))};

%o (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, (d%2) * (-1)^(d\2) * kronecker( n/d, 5)))};

%o (PARI) {a(n) = if( n<1, 0, qfrep( [1, 0; 0, 5], n)[n] - qfrep( [2, 1; 1, 3], n)[n])};

%Y Cf. A033764, A035170, A129391.

%K sign,mult

%O 1,3

%A _Michael Somos_, Aug 22 2005